Let us consider an American call option with strike price K and the time to maturity be T. Assume that the underlying stock does not pay any dividend. Let the price of this call option is C$^a$ today (t = 0). Now, suppose that at some intermediate time t ($<$T), I decide to exercise my call option. Hence the profit is:

P1 = S(t) - K - C$^a$

I could then earn the interest on this profit and hence at maturity i will have:

P2 = P1*e$^{r(T-t)}$ = (S(t) - K - C$^a$)e$^{r(T-t)}$

Instead, I could have waited and exercised it at maturity. My profit would then be:

P3 = S(T) - K + Ke$^{rT}$ - C$^a$

I write this because i could have kept $K in the bank at t = 0 and earned a risk-free interest on it till maturity time T.

So here is my question: Merton (in 1973) said that it an American call on a non-dividend paying stock should not be exercised before expiration. I am just trying to figure out why it is true. Because there might be a possibility that P2 > P3.

P.S: I am not contesting that what Merton said is wrong. I totally respect him and am sure what he is saying is correct. But I am not able to see it mathematically. Any help will be appreciated!.

2 Answers
2

You compare apples and oranges here. You can't possibly compare the profit generated involving S(t) on one side and S(T) on the other side. at time t you do not know what the stock will be worth at time T. Merton made the statement in the context of deciding whether

to exercise the call option at any time before expiration

OR

to simply sell the call option in the market

and came to the conclusion that it is sub-optimal to exercise the option before expiration, but in light of the fact that he meant a comparison between exercise vs. selling the option, not between exercising and waiting till expiration.

A logical way of answering this question is proof by contradiction. First note that if it is not optimal to exercise an american option prior to expiry, then the option should have the same value as the european option.

So assume that it is not optimal to exercise the option prior to expiry, i.e. assume that the american option has the same value as the european option.

Determine the value of the option using the Black-Scholes formula, for a range of initial spot prices, with all other parameters fixed.

Plot a graph of the option values against the spot prices, and on the same chart plot the payoff against the spot prices.

You observe that for a call option with no dividends, for any given spot price, the option value always exceeds the corresponding payoff value. This means that it will always be profitable to sell on, or indeed hold the option, rather than to exercise - i.e. it is never optimal to exercise the option prior to expiry.

On the other hand, where there is a dividend and the spot price is large enough, the payoff is larger than option value - i.e. it is optimal to exercise the option. This is where the initial assumption that it is not optimal to exercise the option prior to expiry is contradicted, in which case we can't hold on to the argument that the European and American call options are of the same value.